Dimension of column- and rowspace

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SUMMARY

The dimension of the row space of a matrix is equal to the dimension of the column space, which is also defined as the rank of the matrix. This relationship holds true for matrices with real elements. Therefore, for any given matrix A, the rank of A corresponds directly to the dimensions of both its row space and column space.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically matrix theory.
  • Familiarity with the definitions of row space and column space.
  • Knowledge of matrix rank and its implications.
  • Basic proficiency in handling matrices with real elements.
NEXT STEPS
  • Study the properties of matrix rank in detail.
  • Explore the relationship between row space and column space in various types of matrices.
  • Learn about the implications of rank in linear transformations.
  • Investigate applications of row and column spaces in solving linear equations.
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators looking to clarify concepts related to matrix dimensions and rank.

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[SOLVED] Dimension of column- and rowspace

Homework Statement


My book doesn't answer this question clearly, but I have notived the following connection:

Does the dimension of the rowspace of a matrix equal the dimension of the columnspace of a matrix which equals the rank of the matrix?
 
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If the matrix has real elements, the row and column spaces have the same dimension. Furthermore, the rank of A equals the dimension of the row space of A.
 

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